An Optimization Approach of Compressive Sensing Recovery Using Split Quadratic Bregman Iteration with Smoothed ℓ0 Norm

Abstract

An optimization algorithm for image recovery is a core issue in the field of compressive sensing (CS). This paper deeply studied the CS reconstruction algorithm based on split Bregman iteration with l 1 norm, which enables the l 1 norm to approximate the original l 0 norm during the optimization process. Consequently, we proposed another novel algorithm improving the precision and the convergence speed based on split quadratic Bregman iteration (SQBI) with l 0 norm. Besides, we analyzed its convergence by proving two monotonically decreasing theorems. Inspired by previous researches, we applied smoothed l 0 norm for the optimization problem to replace the traditional l 0 norm in CS. The improvement is made by using a Gaussian function to approximate the l 0 norm, transforming it into a convex optimization problem, and eventually achieved a convergent solution by the steepest descent method. The experimental results show that under the same conditions, compared with other state-of-the-art algorithms, the reconstruction accuracy of the CS reconstruction algorithm based on the SQBI with smoothed l 0 norm is improved significantly, and its convergence rate is also accelerated as well.